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Catalytic Quantum Error Correction: Theory, Efficient Catalyst Preparation, and Numerical Benchmarks

Hikaru Wakaura

Abstract

We introduce Catalytic Quantum Error Correction (CQEC), a state recovery protocol exploiting catalytic covariant transformations. CQEC recovers a known target state from noisy copies without an error \emph{magnitude} threshold: recovery succeeds whenever the coherent modes satisfy $\mathcal{C}(ρ_0) \subseteq \mathcal{C}(ρ_\mathrm{noisy})$, regardless of noise strength. The main practical bottleneck -- catalyst preparation requiring $n^* \sim d^4 e^{2γ}$ copies -- is resolved by a three-stage pipeline combining CPMG dynamical decoupling, Clifford twirling, and the recursive swap test, achieving $F_\mathrm{cat} > 0.96$ with only 8~copies ($10^9$-fold reduction). Numerical validation across four quantum algorithms ($d = 4$--$64$), a cryptographic protocol, and three noise models confirms $F > 0.999$ in the asymptotic limit across 200~configurations.

Catalytic Quantum Error Correction: Theory, Efficient Catalyst Preparation, and Numerical Benchmarks

Abstract

We introduce Catalytic Quantum Error Correction (CQEC), a state recovery protocol exploiting catalytic covariant transformations. CQEC recovers a known target state from noisy copies without an error \emph{magnitude} threshold: recovery succeeds whenever the coherent modes satisfy , regardless of noise strength. The main practical bottleneck -- catalyst preparation requiring copies -- is resolved by a three-stage pipeline combining CPMG dynamical decoupling, Clifford twirling, and the recursive swap test, achieving with only 8~copies (-fold reduction). Numerical validation across four quantum algorithms (--), a cryptographic protocol, and three noise models confirms in the asymptotic limit across 200~configurations.

Paper Structure

This paper contains 39 sections, 24 equations, 13 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Unspeakable coherence and covariant operations
  4. Coherent modes and mode inclusion
  5. CQEC protocol
  6. Copy scaling
  7. Quantum Circuit Architecture
  8. Energy-conserving gate
  9. Three-layer circuit structure
  10. Minimal 4-qubit implementation
  11. Parameter optimization
  12. Efficient Catalyst Preparation
  13. Resource bottleneck
  14. Variational catalyst preparation
  15. Recursive swap test purification
  16. ...and 24 more sections

Figures (13)

  • Figure 1: Minimal 4-qubit CQEC circuit with 5 EC gates [Eq. \ref{['eq:ec_gate_matrix']}]. Vertical lines connect the two qubits of each gate. Colors denote layers: L1 (S$\leftrightarrow$C), L2 (C$\leftrightarrow$A), L3 (S$\leftrightarrow$A).
  • Figure 2: Pipeline comparison under dephasing $\gamma = 2$ with $n = 8$ copies (CPMG-8). Raw purification (red) fails for $d \geq 8$. DD alone (orange) improves but is dimension-dependent. Twirl alone (purple) fails. DD+Twirl (green) achieves $F_\mathrm{cat} > 0.96$ uniformly.
  • Figure 3: Catalyst fidelity vs. CPMG pulse count for qDRIFT ($d = 8$, $n = 8$ copies, $\gamma = 2$). DD reduces effective dephasing, twirling isotropizes, and the swap test purifies efficiently.
  • Figure 4: Summary of all catalyst preparation strategies under dephasing $\gamma = 2$: variational (0 copies), standard swap test, covariant swap test, and DD+Twirl+Swap Test pipeline (CPMG-8, 8 copies). The DD+Twirl pipeline dramatically outperforms all other methods.
  • Figure 5: Sharp threshold of CQEC. Post-correction fidelity vs. residual coherence $\varepsilon$ (log scale). Green circles: successful recovery ($\varepsilon > 0$). Red cross: failed recovery ($\varepsilon = 0$). Dashed line: $F = 1/d = 0.25$.
  • ...and 8 more figures